Numerical Methods for Parabolic PDEs

The heat/diffusion equation is presented as our primary example of a parabolic PDE. Conservative methods are developed for the linear and nonlinear heat/diffusion equations expressed as parabolic conservation laws. Our standard timestepping methods for ODE initial value problems all work for the diffusion equation (with second-order accurate central differences for spatial derivatives): forward Euler, backward Euler, TR, and TRBDF2—highly recommended, since the diffusion equation is stiff. To analyze the stability of numerical methods for time-dependent PDEs, von Neumann’s Fourier analysis of the numerical growth factor is introduced. For parabolic PDEs, the timestep is adjusted dynamically based on an estimate of the local error or from a divided-difference formula for TRBDF2. TRBDF2 and Newton’s method are applied to simulating nonlinear diffusion for semiconductor wafer processing.